Integrals with inverse trig, long division and completing the square

Answer Key - Grade 12 Math Worksheet - Challenge Me

Circle the best answer, or write your answer on the line.

1. 1.

   Esme substitutes and uses an inverse trig rule to find ∫ cos x/(1 + sin² x) dx. Which is it?

   • ln(1 + sin² x) + C
   • arctan(sin x) + C
   • arctan(cos x) + C
2. 2.

   Jun substitutes and uses an inverse trig rule to find ∫ 1/(x√(1 − (ln x)²)) dx. Which is it?

   • ln(1 − (ln x)²) + C
   • arcsin(x) + C
   • arcsin(ln x) + C
3. 3.

   Kira divides first to evaluate ∫ from 1 to 4 of (x² + 3)/(x + 1) dx. What is it, to the nearest hundredth?

   8.17
4. 4.

   Luz divides first to evaluate ∫ from 0 to 3 of (x² + 5)/(x + 2) dx. What is it, to the nearest hundredth?

   6.75
5. 5.

   Milo divides first to evaluate ∫ from 1 to 3 of (2x² + 1)/(x + 1) dx. What is it, to the nearest hundredth?

   6.08
6. 6.

   Noah divides first to evaluate ∫ from 0 to 1 of (x³ + 2)/(x + 1) dx. What is it, to the nearest hundredth?

   1.53
7. 7.

   Pia decides between a log and an arctan for ∫ x/(x² + 9) dx. Which is the antiderivative?

   • (1/3) arctan(x/3) + C
   • (1/2) ln(x² + 9) + C
   • ln(x² + 9) + C
8. 8.

   Ren decides between a log and an arctan for ∫ 1/(x² + 9) dx. Which is the antiderivative?

   • (1/2) ln(x² + 9) + C
   • (1/3) arctan(x/3) + C
   • arctan(x/3) + C
9. 9.

   Tala decides between a log and an arctan for ∫ 2x/(x² + 16) dx. Which is the antiderivative?

   • ln(x² + 16) + C
   • arctan(x/4) + C
   • 2 ln(x² + 16) + C
10. 10.

    Zee decides between a log and an arctan for ∫ 3/(x² + 4) dx. Which is the antiderivative?

    • (3/2) ln(x² + 4) + C
    • 3 arctan(x/2) + C
    • (3/2) arctan(x/2) + C